A Novel Method to Generate Key-Dependent S-Boxes with Identical Algebraic Properties

Ahmad Y. Al-Dweik; Iqtadar Hussain; Moutaz S. Saleh; M. T. Mustafa

REVIEW 2 major objections 3 minor 38 references

A Novel Method to Generate Key-Dependent S-Boxes with Identical Algebraic Properties

T0 review · 2 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Two key-derived permutations applied to an S-box's rows and columns produce clone S-boxes whose bijection, nonlinearity, SAC, and BIC are identical to the seed.

desk verdict Sound preservation theorem, but the claimed (n!)^2 distinct clone S-boxes is unproved and false in general; the paper needs a corrected key-space bound before it is publishable. read the letter →

arxiv 1908.09168 v2 pith:R7FQXW7E submitted 2019-08-24 cs.CR math.GR

Ahmad Y. Al-Dweik , Iqtadar Hussain , Moutaz S. Saleh , M. T. Mustafa This is my paper · ORCID

classification cs.CRmath.GR MSC 94A6006E3020B35

keywords CryptographyKey-Dependents-boxesPermutationBijectionNonlinearityStrictavalanchecriterionBitsindependenceInvariant

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Dynamic S-boxes improve a cipher's resistance to linear and differential attacks, but earlier key-dependent designs do not keep nonlinearity invariant. This paper claims to close that gap: apply one permutation to the columns and another to the rows of an S-box's bit matrix, and the result is a new S-box whose bijection, nonlinearity, strict avalanche criterion (SAC), and bit independence criterion (BIC) are identical to the seed. The construction is algorithmic, with the two permutations extracted from the key, and a proof of invariance is given for all four metrics. If the claim is correct, a designer can take one strong S-box, such as the AES S-box, and generate many key-dependent variants without re-running expensive metric evaluations.

What carries the argument

The load-bearing object is the transformation Q1 Y P2, assembled from the identity S-box X and two key-derived permutation matrices P1 and P2, with Q1 the 2^n×2^n row-permutation matrix that satisfies Q1 X = X P1. This identity is what turns an abstract row permutation of the S-box's bit matrix into a concrete permutation of the input coordinates, x ↦ x P1, so the known invariance of nonlinearity and SAC under nonsingular affine input changes transfers directly to the clone. The group action is S_n on the columns and the induced image of S_n inside S_{2^n} on the rows, which is why the paper counts (n!)^2 candidate clones.

What would settle it

Enumerate all (n!)^2 pairs for the n=4 seed S-box from Example 4.1 and compare the resulting vectors; if two different pairs produce the same S-box, the claimed clone count is false. The same enumeration on a randomly chosen n=5 or n=6 seed would also test whether the metric-preservation theorem holds outside the two examples.

Watch

Extended reading notes

Core claim

The central claim, Theorem 3.3, is that for any n×n S-box Y and any two n×n permutation matrices P1,P2, the 2^n×n bit matrix Q1 Y P2 is an S-box with the same bijection, nonlinearity, SAC, and BIC as Y, where Q1 is the 2^n×2^n permutation matrix determined by the identity S-box X through the identity Q1 X = X P1. The column factor P2 simply relabels the output bits, so it plainly preserves all four metrics; the row factor Q1 is the subtle part. Because Q1 X = X P1, permuting the rows of the truth-table matrix is the same as permuting the coordinates of the input x to x P1, and the standard lemmas on affine input changes carry the nonlinearity and SAC invariance, while applying the same lemmas to fj⊕fk carries BIC. The paper verifies the construction on the n=4 seed S-box used in Example 4.1 for all 576 permutation pairs and on the n=8 AES S-box for one pair, reporting metric tables that match the seeds exactly.

Load-bearing premise

The paper's advertised key space of (n!)^2 distinct clones assumes that different key-derived permutation pairs always produce different S-boxes; this injectivity is stated but not proven, and a collision would shrink the effective key space.

Editorial extensions

If this is right

A single strong static S-box can be expanded into a large family of key-dependent S-boxes whose four standard metrics are exactly those of the seed, so designers do not need to regenerate and re-test S-boxes for each key.
For an n×n S-box the construction yields (n!)^2 clones in the paper's count; for n=8 this is (8!)^2 ≈ 1.6×10^9 candidates from one AES seed.
If the seed has fixed points or reverse fixed points, the algorithm's update step produces clone S-boxes without those points while preserving the four metrics, addressing the exploitable weakness discussed in the paper's Sec. 4.
The method directly answers the earlier finding that key-dependent Rijndael variants do not keep nonlinearity invariant: the generated clones preserve nonlinearity exactly.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The row permutation Q1 is not an arbitrary member of S_{2^n}: it is the image of S_n acting on input coordinates through the identity S-box. The family generated is therefore an orbit of the seed under a group isomorphic to S_n × S_n, a subgroup of the affine group, not under all input/output permutations.
The claimed (n!)^2 distinct clones presupposes that different pairs (P1,P2) give different S-boxes; the paper reports checking this for the n=4 example, but a general injectivity proof is not given, so the effective key space could be smaller for symmetric seeds.
Because the clone is related to the seed by an input coordinate permutation and an output bit permutation, the construction is metric-preserving but not metric-improving: it cannot raise nonlinearity above the seed's value, only produce many equally strong variants.
A natural stress test is to apply the construction to random S-boxes of size n=5 or n=6 and compare full nonlinearity, SAC, and BIC tables; if invariance ever fails outside the two worked examples, the theorem's scope would need qualification.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The core of this paper is a correct and pleasantly simple observation: if you permute the input bits and output bits of an S-box in the specific way they define, the four classical metrics—bijection, nonlinearity, SAC, and BIC—are preserved. The proof via the identity-sbox relation Q1X = XP1 is tidy, and the lemmas from Pieprzyk et al. transfer cleanly. For someone who wants key-dependent S-boxes without degrading these properties, this gives a cheap way to generate many candidates from a trusted seed. The examples for n=4 and n=8 check out, and the tables match.

The soft spot is the headline count. The abstract and Remark 4.2 claim (n!)^2 distinct clones, but the paper never proves injectivity of the map (σ1,σ2) → Q1YP2. That map has collisions whenever the seed has a nontrivial coordinate-permutation stabilizer. The identity seed is a concrete counterexample: Q1XP2 = XP1P2, so you only get n! distinct outputs, not (n!)^2. The authors verify 576 distinct outputs for the 4×4 Adams–Tavares seed, but that is a check, not a proof, and for the AES seed no such check is reported. Since the effective key space is the main selling point, this is a real flaw, not a minor omission. They should either prove a stabilizer bound or lower the claim to "at most (n!)^2, typically much fewer."

The fixed-point-removal steps (10–11) are also asserted without proof. I don't doubt they can be made to work, but as written there is no termination argument and no proof that the updated permutations preserve the claimed properties. That section needs a rewrite or a clear statement of scope.

The Maple code has minor issues (the BLmatrix procedure uses convert base 10 to 2, which is not correct for all inputs), but that is cosmetic.

Overall: the central preservation theorem is sound and useful as a practical recipe, but the paper overstates the key space. With a corrected count and a proper treatment of the fixed-point loop, it would be publishable. I'd send it to a referee; the referee should ask for these changes, not for a rejection.

Referee Report

2 major / 3 minor

Summary. The paper proposes a method for generating key-dependent S-boxes, called clone S-boxes, by applying two permutations to an initial seed S-box: one permutation acts on the input-coordinate bits and is realized as a row permutation Q1 through the identity S-box X, and the second permutation P2 permutes the output columns. The central claim, Theorem 3.3, is that Q1 Y P2 is a new S-box with the same algebraic properties as the seed Y: bijection, nonlinearity, the strict avalanche criterion (SAC), and the bits independence criterion (BIC). The proof argues that row permutation by Q1 is equivalent to an input coordinate permutation and that the cited invariance lemmas transfer nonlinearity and SAC, while bijection and BIC are preserved. The paper includes a Maple implementation, worked examples for n=4 and n=8, and tables comparing the four metrics of the seed and clone S-boxes. The abstract and Section 4 further claim that the method generates (4!)^2 distinct clones for the n=4 seed and, for n=8, 8! clones per column permutation, and that fixed points and reverse fixed points can be removed while preserving the algebraic properties.

Significance. If the main claims hold, the paper offers a simple and computationally cheap method for producing dynamic key-dependent S-boxes whose nonlinearity, SAC, and BIC exactly match a strong seed such as the AES S-box, directly addressing the concern in [2] that key-dependent variations degrade nonlinearity. The invariance proof in Theorem 3.3 is mathematically sound and rests on standard, correctly cited lemmas; there is no parameter fitting or circularity. The reproduction of the reported metric tables for the two examples is a useful sanity check. However, two load-bearing claims are not established: the claimed count of (n!)^2 distinct clone S-boxes, and the guaranteed removal of fixed points by the loop in steps 10-11. Since the practical motivation is a large key-dependent space of distinct S-boxes, the unproved distinctness assertion limits the paper's significance as it stands.

major comments (2)

[Abstract and Remark 4.2] The claim that the method generates (4!)^2 = 576 distinct S-boxes for the n=4 seed, and by extension (n!)^2 clones for n=8, requires injectivity of the map (P1,P2) → Q1 Y P2. Injectivity is asserted but never proved, and it is false in general. For the identity seed Y = X, Q1 X P2 = X P1 P2, so the output depends only on the product P1P2; there are n! distinct clones, not (n!)^2. The authors verified 576 outputs for the particular 4×4 seed, but no such verification is reported for the AES seed, and no stabilizer argument is given to rule out collisions for arbitrary seeds. Because the size of the effective key space is the main practical selling point, this gap must be fixed, either by proving the count under explicit conditions on Y or by revising the claims to state the actual number of distinct clones for the presented seeds.
[Algorithm steps 10-11 and Section 4 opening paragraph] The fixed-point and reverse-fixed-point removal loop is not justified. The paper asserts that if the initial S-box has fixed points, 'the algorithm can be applied to obtain improved clone versions where all the fixed points and reverse fixed points are removed,' but no proof is given that for an arbitrary bijective S-box Y there exist permutations σ̄1, σ̄2 such that the resulting clone has no fixed points, nor is there a termination argument for the loop. The set of allowed row permutations is restricted to bit-coordinate permutations, not all permutations of {0,1}^n, so the existence of a derangement in the full symmetric group does not by itself imply the existence of a fixed-point-free clone in this restricted class. The authors should either prove the existence, provide a concrete termination bound, or state the condition under which the loop is guaranteed to succeed.

minor comments (3)

[Abstract and page 2] There are several wording and typographical issues: 'To generated (4!)^2 clone copies' should read 'To generate', and the phrase 'initial n×n s-box' should be 'initial n-bit s-box' or '2^n×n s-box' to avoid confusing the S-box size with the permutation size.
[Appendix, Listing 5] The Maple procedure FIXP hardcodes the reverse fixed-point condition as 'S[i]=255-(i-1)', which is specific to n=8; for general n this should be 2^n-1-(i-1) or should be parameterized by n. The main text also never defines 'reverse fixed point' before using it in step 10.
[Section 4 and Example 4.3] The phrase 'Simillarly' in Example 4.3 is a typo, and the text would benefit from a statement of whether the AES-seed clone was checked for distinctness against all other generated clones, since Remark 4.2 reports such a check only for the n=4 seed.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the preservation proof is self-contained and relies on external textbook lemmas; the unproved (n!)^2 distinctness count is a correctness concern, not a circular reduction.

full rationale

The paper's central claim is that Q1YP2 preserves bijection, nonlinearity, SAC, and BIC relative to the seed Y. The proof does not fit any parameter and does not assume the target result. Q1 is explicitly constructed from P1 via Lemma 3.2 so that Q1X = XP1; the theorem then identifies the row-permuted s-box with the input-coordinate-permuted Boolean functions. Nonlinearity invariance follows from Lemma 2.4, and SAC/BIC follow from Theorem 2.5, both cited to the external textbook [36]; bijection follows from Proposition 2.1 cited to Adams and Tavares [7]. These are standard, independent results whose assumptions do not include the preservation claim. Self-citations to [3,4,5,6] appear only as background for extending column-only permutation to row-and-column permutation and are not load-bearing; no uniqueness theorem is invoked, and no ansatz is smuggled in by self-citation. The main weakness flagged by the skeptical reading is the asserted count of (n!)^2 distinct clone s-boxes, which is stated in the abstract, Remark 4.2, and the discussion before Example 4.3 but never proved; seeds with coordinate-permutation symmetries can cause collisions. That is a mathematical correctness or proof-gap issue, not circularity: the construction still produces valid clone s-boxes, and the preservation theorem does not depend on injectivity of the map (P1,P2) -> Q1YP2. Accordingly, under the hard rules requiring an exhibited reduction or fitted-input-renamed-as-prediction, no circular step can be substantiated, and the appropriate score is 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper introduces no new constants or entities. The core theorem borrows two standard invariance results from [36]. The unproven extras are the distinctness count and the guaranteed fixed-point removal.

assumptions (4)

standard math Nonlinearity is invariant under nonsingular affine input transformations, and SAC is preserved under input coordinate permutations (Lemma 2.4 and Theorem 2.5 from [36]).
The clone preservation proof uses these external results directly and does not re-derive them.
domain assumption The row permutation Q1 equals an input coordinate permutation x to xP1, so row permuting Y is equivalent to substituting xP1 into each Boolean component.
Equation (3.2) relies on X being the identity S-box and on Lemma 3.2; this identification is valid for the identity seed but is the bridge that makes the invariance lemmas applicable.
ad hoc to paper For any seed S-box with fixed points, there exists a permutation pair whose clone has no fixed points, and the loop in steps 10-11 finds it.
Section 4 asserts improved clones with fixed points removed, but no construction, choice rule for sigma-bar, or termination proof is given.
ad hoc to paper The permutation-to-clone map is injective, so (n!)^2 distinct clones exist.
The abstract and Remark 4.2 count (4!)^2 and (8!)^2 clones without proving distinctness.

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Pith. "Pith review of A Novel Method to Generate Key-Dependent S-Boxes with Identical Algebraic Properties." pith.science (2026). https://pith.science/paper/R7FQXW7E

@misc{pith2026190809168,
  author       = {Pith},
  title        = {Pith review of: A Novel Method to Generate Key-Dependent S-Boxes with Identical Algebraic Properties},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/R7FQXW7E}},
  note         = {Machine review of arXiv:1908.09168}
}

abstract

The s-box plays the vital role of creating confusion between the ciphertext and secret key in any cryptosystem, and is the only nonlinear component in many block ciphers. Dynamic s-boxes, as compared to static, improve entropy of the system, hence leading to better resistance against linear and differential attacks. It was shown in [2] that while incorporating dynamic s-boxes in cryptosystems is sufficiently secure, they do not keep non-linearity invariant. This work provides an algorithmic scheme to generate key-dependent dynamic $n\times n$ clone s-boxes having the same algebraic properties namely bijection, nonlinearity, the strict avalanche criterion (SAC), the output bits independence criterion (BIC) as of the initial seed s-box. The method is based on group action of symmetric group $S_n$ and a subgroup $S_{2^n}$ respectively on columns and rows of Boolean functions ($GF(2^n)\to GF(2)$) of s-box. Invariance of the bijection, nonlinearity, SAC, and BIC for the generated clone copies is proved. As illustration, examples are provided for $n=8$ and $n=4$ along with comparison of the algebraic properties of the clone and initial seed s-box. The proposed method is an extension of [3,4,5,6] which involved group action of $S_8$ only on columns of Boolean functions ($GF(2^8)\to GF(2)$ ) of s-box. For $n=4$, we have used an initial $4\times 4$ s-box constructed by Carlisle Adams and Stafford Tavares [7] to generated $(4!)^2$ clone copies. For $n=8$, it can be seen [3,4,5,6] that the number of clone copies that can be constructed by permuting the columns is $8!$. For each column permutation, the proposed method enables to generate $8!$ clone copies by permuting the rows.

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Pith tools

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[1] [2]

An Examination of Ineﬃciencies in Key Dependent Vari- ations of the Rijndael S-Box

Easttom C. An Examination of Ineﬃciencies in Key Dependent Vari- ations of the Rijndael S-Box. InElectrical Engineering (ICEE), Iranian Conference on 2018 May 8 (pp. 1658-1663). IEEE

work page 2018

[2] [1]

”Cryptanalysis and constructing S-Box based on chaotic map and backtracking.” Ap- plied Mathematics and Computation 376 (2020): 125153

Liu, Hongjun, Abdurahman Kadir, and Chengbo Xu. ”Cryptanalysis and constructing S-Box based on chaotic map and backtracking.” Ap- plied Mathematics and Computation 376 (2020): 125153

work page 2020

[3] [3]

Construction of S8 Liu J S-boxes and their applications

Iqtadar Hussain, Tariq Shah Hasan Mahmood, Muhammad Asif Gondal. Construction of S8 Liu J S-boxes and their applications. Computers and Mathematics with Applications 64 (2012) 2450–2458

work page 2012

[4] [4]

S8 aﬃne-power-aﬃne S-boxes and their applications

Iqtadar Hussain, Tariq Shah, Muhammad Asif Gondal, Hasan Mah- mood. S8 aﬃne-power-aﬃne S-boxes and their applications. Neural Comput and Applic (2012) 21 (Suppl 1):S377–S383

work page 2012

[5] [5]

A noise resistant symmetric key cryptosystem based on S8 S- boxes and chaotic maps

Iqtadar Hussain, Amir Anees, Muhammad Aslam, Rehan Ahmed, Nasir Siddiqui. A noise resistant symmetric key cryptosystem based on S8 S- boxes and chaotic maps. The European Physical Journal Plus, April 2018, pp.133:167

work page 2018

[6] [6]

Neural Computing and Applications

Designing secure substitution boxes based on permutation of symmet- ric group Amir Anees, Yi-Ping Phoebe Chen. Neural Computing and Applications. https://doi.org/10.1007/s00521-019-04207-8 17

work page doi:10.1007/s00521-019-04207-8

[7] [7]

The Structured Design of Crypto- graphically Good S-Boxes, Journal of Cryptology, 3 (1), pp

Adms, C.M., Tavares, S.E., 1990. The Structured Design of Crypto- graphically Good S-Boxes, Journal of Cryptology, 3 (1), pp. 27-41

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